One option would be to create a Page dedicated to processing these GET requests. Something like http://example.com/create/?yourquerystringhere`.
You can then create a page template, either page-create.php to automatically apply to that page, or a custom template you manually apply from the dropdown. That page template can contain your custom code:
<?php
// if any variables are missing, redirect to homepage
if(empty($_GET['user_id']) || empty($_GET['entry_id']) || empty($_GET['nonce'])) {
wp_redirect();
// otherwise, process the request
} else {
// add your code here to make sure user_id, entry_id, and nonce are valid
// if they're valid, publish the post - you can then display a success
// message, or redirect to the newly published post
// you should also check whether entry_id is already published
// and if so, auto redirect to that rather than re-publishing
}
?>
This way you’re just creating a Page template (and a child theme if you don’t already have a child theme or a custom one) and one Page to handle all the processing.
Another option would be to set this type of code in your child theme’s front-page.php so you don’t have to create a Page at all, and if the GET vars are empty or invalid, carry on displaying whatever normally goes on the homepage, else process as above.